3.49 \(\int (c+d x) (a+b \cot (e+f x))^3 \, dx\)

Optimal. Leaf size=278 \[ -\frac{3 i a^2 b d \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac{i b^3 d \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a b^2 (c+d x) \cot (e+f x)}{f}-3 a b^2 c x+\frac{3 a b^2 d \log (\sin (e+f x))}{f^2}-\frac{3}{2} a b^2 d x^2-\frac{b^3 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x) \cot ^2(e+f x)}{2 f}+\frac{i b^3 (c+d x)^2}{2 d}-\frac{b^3 d \cot (e+f x)}{2 f^2}-\frac{b^3 d x}{2 f} \]

[Out]

-3*a*b^2*c*x - (b^3*d*x)/(2*f) - (3*a*b^2*d*x^2)/2 + (a^3*(c + d*x)^2)/(2*d) - (((3*I)/2)*a^2*b*(c + d*x)^2)/d
+ ((I/2)*b^3*(c + d*x)^2)/d - (b^3*d*Cot[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)*Cot[e + f*x])/f - (b^3*(c + d
*x)*Cot[e + f*x]^2)/(2*f) + (3*a^2*b*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f - (b^3*(c + d*x)*Log[1 - E^((2*
I)*(e + f*x))])/f + (3*a*b^2*d*Log[Sin[e + f*x]])/f^2 - (((3*I)/2)*a^2*b*d*PolyLog[2, E^((2*I)*(e + f*x))])/f^
2 + ((I/2)*b^3*d*PolyLog[2, E^((2*I)*(e + f*x))])/f^2

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Rubi [A]  time = 0.322774, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3722, 3717, 2190, 2279, 2391, 3720, 3475, 3473, 8} \[ -\frac{3 i a^2 b d \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac{i b^3 d \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 a b^2 (c+d x) \cot (e+f x)}{f}-3 a b^2 c x+\frac{3 a b^2 d \log (\sin (e+f x))}{f^2}-\frac{3}{2} a b^2 d x^2-\frac{b^3 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x) \cot ^2(e+f x)}{2 f}+\frac{i b^3 (c+d x)^2}{2 d}-\frac{b^3 d \cot (e+f x)}{2 f^2}-\frac{b^3 d x}{2 f} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*(a + b*Cot[e + f*x])^3,x]

[Out]

-3*a*b^2*c*x - (b^3*d*x)/(2*f) - (3*a*b^2*d*x^2)/2 + (a^3*(c + d*x)^2)/(2*d) - (((3*I)/2)*a^2*b*(c + d*x)^2)/d
+ ((I/2)*b^3*(c + d*x)^2)/d - (b^3*d*Cot[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)*Cot[e + f*x])/f - (b^3*(c + d
*x)*Cot[e + f*x]^2)/(2*f) + (3*a^2*b*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f - (b^3*(c + d*x)*Log[1 - E^((2*
I)*(e + f*x))])/f + (3*a*b^2*d*Log[Sin[e + f*x]])/f^2 - (((3*I)/2)*a^2*b*d*PolyLog[2, E^((2*I)*(e + f*x))])/f^
2 + ((I/2)*b^3*d*PolyLog[2, E^((2*I)*(e + f*x))])/f^2

Rule 3722

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (c+d x) (a+b \cot (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)+3 a^2 b (c+d x) \cot (e+f x)+3 a b^2 (c+d x) \cot ^2(e+f x)+b^3 (c+d x) \cot ^3(e+f x)\right ) \, dx\\ &=\frac{a^3 (c+d x)^2}{2 d}+\left (3 a^2 b\right ) \int (c+d x) \cot (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x) \cot ^2(e+f x) \, dx+b^3 \int (c+d x) \cot ^3(e+f x) \, dx\\ &=\frac{a^3 (c+d x)^2}{2 d}-\frac{3 i a^2 b (c+d x)^2}{2 d}-\frac{3 a b^2 (c+d x) \cot (e+f x)}{f}-\frac{b^3 (c+d x) \cot ^2(e+f x)}{2 f}-\left (6 i a^2 b\right ) \int \frac{e^{2 i (e+f x)} (c+d x)}{1-e^{2 i (e+f x)}} \, dx-\left (3 a b^2\right ) \int (c+d x) \, dx-b^3 \int (c+d x) \cot (e+f x) \, dx+\frac{\left (3 a b^2 d\right ) \int \cot (e+f x) \, dx}{f}+\frac{\left (b^3 d\right ) \int \cot ^2(e+f x) \, dx}{2 f}\\ &=-3 a b^2 c x-\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{i b^3 (c+d x)^2}{2 d}-\frac{b^3 d \cot (e+f x)}{2 f^2}-\frac{3 a b^2 (c+d x) \cot (e+f x)}{f}-\frac{b^3 (c+d x) \cot ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\sin (e+f x))}{f^2}+\left (2 i b^3\right ) \int \frac{e^{2 i (e+f x)} (c+d x)}{1-e^{2 i (e+f x)}} \, dx-\frac{\left (3 a^2 b d\right ) \int \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}-\frac{\left (b^3 d\right ) \int 1 \, dx}{2 f}\\ &=-3 a b^2 c x-\frac{b^3 d x}{2 f}-\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{i b^3 (c+d x)^2}{2 d}-\frac{b^3 d \cot (e+f x)}{2 f^2}-\frac{3 a b^2 (c+d x) \cot (e+f x)}{f}-\frac{b^3 (c+d x) \cot ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\sin (e+f x))}{f^2}+\frac{\left (3 i a^2 b d\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^2}+\frac{\left (b^3 d\right ) \int \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=-3 a b^2 c x-\frac{b^3 d x}{2 f}-\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{i b^3 (c+d x)^2}{2 d}-\frac{b^3 d \cot (e+f x)}{2 f^2}-\frac{3 a b^2 (c+d x) \cot (e+f x)}{f}-\frac{b^3 (c+d x) \cot ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\sin (e+f x))}{f^2}-\frac{3 i a^2 b d \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}-\frac{\left (i b^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^2}\\ &=-3 a b^2 c x-\frac{b^3 d x}{2 f}-\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}-\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{i b^3 (c+d x)^2}{2 d}-\frac{b^3 d \cot (e+f x)}{2 f^2}-\frac{3 a b^2 (c+d x) \cot (e+f x)}{f}-\frac{b^3 (c+d x) \cot ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\sin (e+f x))}{f^2}-\frac{3 i a^2 b d \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{i b^3 d \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}\\ \end{align*}

Mathematica [A]  time = 3.52256, size = 276, normalized size = 0.99 \[ \frac{\sin (e+f x) (a+b \cot (e+f x))^3 \left (i b d \left (b^2-3 a^2\right ) \sin ^2(e+f x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )+\sin ^2(e+f x) \left (-(e+f x) \left (3 i a^2 b d (e+f x)+a^3 (d (e-f x)-2 c f)+3 a b^2 (2 c f-d e+d f x)-i b^3 d (e+f x)\right )+2 b \log (\sin (e+f x)) \left (a^2 (3 c f-3 d e)+3 a b d+b^2 (d e-c f)\right )-2 b d \left (b^2-3 a^2\right ) (e+f x) \log \left (1-e^{2 i (e+f x)}\right )\right )-\frac{1}{2} b^2 (\sin (2 (e+f x)) (6 a f (c+d x)+b d)+2 b f (c+d x))\right )}{2 f^2 (a \sin (e+f x)+b \cos (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)*(a + b*Cot[e + f*x])^3,x]

[Out]

((a + b*Cot[e + f*x])^3*Sin[e + f*x]*((-((e + f*x)*((3*I)*a^2*b*d*(e + f*x) - I*b^3*d*(e + f*x) + 3*a*b^2*(-(d
*e) + 2*c*f + d*f*x) + a^3*(-2*c*f + d*(e - f*x)))) - 2*b*(-3*a^2 + b^2)*d*(e + f*x)*Log[1 - E^((2*I)*(e + f*x
))] + 2*b*(3*a*b*d + b^2*(d*e - c*f) + a^2*(-3*d*e + 3*c*f))*Log[Sin[e + f*x]])*Sin[e + f*x]^2 + I*b*(-3*a^2 +
b^2)*d*PolyLog[2, E^((2*I)*(e + f*x))]*Sin[e + f*x]^2 - (b^2*(2*b*f*(c + d*x) + (b*d + 6*a*f*(c + d*x))*Sin[2
*(e + f*x)]))/2))/(2*f^2*(b*Cos[e + f*x] + a*Sin[e + f*x])^3)

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Maple [B]  time = 0.47, size = 745, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)*(a+b*cot(f*x+e))^3,x)

[Out]

b^3/f^2*d*e*ln(exp(I*(f*x+e))-1)-6*b/f*a^2*c*ln(exp(I*(f*x+e)))-6*b^2/f^2*a*d*ln(exp(I*(f*x+e)))-2*b^3/f^2*d*e
*ln(exp(I*(f*x+e)))+I*b^3/f^2*d*polylog(2,-exp(I*(f*x+e)))-3/2*I*a^2*b*d*x^2+1/2*a^3*d*x^2+a^3*c*x+1/2*I*b^3*d
*x^2-I*b^3*c*x+I*b^3/f^2*d*polylog(2,exp(I*(f*x+e)))+3*b/f*a^2*c*ln(exp(I*(f*x+e))-1)+I*b^3/f^2*d*e^2-3*a*b^2*
c*x-3/2*a*b^2*d*x^2+3*I*a^2*b*c*x+b^2*(-6*I*a*d*f*x*exp(2*I*(f*x+e))-6*I*a*c*f*exp(2*I*(f*x+e))+2*b*d*f*x*exp(
2*I*(f*x+e))+6*I*a*d*f*x-I*b*d*exp(2*I*(f*x+e))+2*b*c*f*exp(2*I*(f*x+e))+6*I*a*c*f+I*b*d)/f^2/(exp(2*I*(f*x+e)
)-1)^2-3*b/f^2*a^2*d*e*ln(exp(I*(f*x+e))-1)+6*b/f^2*a^2*d*e*ln(exp(I*(f*x+e)))+3*b/f^2*ln(1-exp(I*(f*x+e)))*a^
2*d*e+3*b/f*ln(exp(I*(f*x+e))+1)*a^2*d*x+3*b/f*ln(1-exp(I*(f*x+e)))*a^2*d*x-3*I*b/f^2*a^2*d*e^2+2*I*b^3/f*d*e*
x-3*I*b/f^2*a^2*d*polylog(2,-exp(I*(f*x+e)))-3*I*b/f^2*a^2*d*polylog(2,exp(I*(f*x+e)))-6*I*b/f*a^2*d*e*x-b^3/f
*ln(1-exp(I*(f*x+e)))*d*x-b^3/f*ln(exp(I*(f*x+e))+1)*d*x-b^3/f^2*ln(1-exp(I*(f*x+e)))*d*e+3*b^2/f^2*a*d*ln(exp
(I*(f*x+e))+1)+3*b^2/f^2*a*d*ln(exp(I*(f*x+e))-1)+3*b/f*a^2*c*ln(exp(I*(f*x+e))+1)-b^3/f*c*ln(exp(I*(f*x+e))-1
)-b^3/f*c*ln(exp(I*(f*x+e))+1)+2*b^3/f*c*ln(exp(I*(f*x+e)))

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Maxima [B]  time = 3.80849, size = 2730, normalized size = 9.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*(a+b*cot(f*x+e))^3,x, algorithm="maxima")

[Out]

1/2*(2*(f*x + e)*a^3*c + (f*x + e)^2*a^3*d/f - 2*(f*x + e)*a^3*d*e/f + 6*a^2*b*c*log(sin(f*x + e)) - 6*a^2*b*d
*e*log(sin(f*x + e))/f - 2*(12*a*b^2*d*e - 12*a*b^2*c*f + (3*a^2*b - 3*I*a*b^2 - b^3)*(f*x + e)^2*d - 2*b^3*d
+ ((6*I*a*b^2 + 2*b^3)*d*e + (-6*I*a*b^2 - 2*b^3)*c*f)*(f*x + e) - (2*b^3*d*e - 2*b^3*c*f + 6*a*b^2*d + 2*(3*a
^2*b - b^3)*(f*x + e)*d + 2*(b^3*d*e - b^3*c*f + 3*a*b^2*d + (3*a^2*b - b^3)*(f*x + e)*d)*cos(4*f*x + 4*e) - 4
*(b^3*d*e - b^3*c*f + 3*a*b^2*d + (3*a^2*b - b^3)*(f*x + e)*d)*cos(2*f*x + 2*e) - (-2*I*b^3*d*e + 2*I*b^3*c*f
- 6*I*a*b^2*d + (-6*I*a^2*b + 2*I*b^3)*(f*x + e)*d)*sin(4*f*x + 4*e) - (4*I*b^3*d*e - 4*I*b^3*c*f + 12*I*a*b^2
*d + (12*I*a^2*b - 4*I*b^3)*(f*x + e)*d)*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) - (2*b^3*d*
e - 2*b^3*c*f + 6*a*b^2*d + 2*(b^3*d*e - b^3*c*f + 3*a*b^2*d)*cos(4*f*x + 4*e) - 4*(b^3*d*e - b^3*c*f + 3*a*b^
2*d)*cos(2*f*x + 2*e) - (-2*I*b^3*d*e + 2*I*b^3*c*f - 6*I*a*b^2*d)*sin(4*f*x + 4*e) - (4*I*b^3*d*e - 4*I*b^3*c
*f + 12*I*a*b^2*d)*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) - 1) + (2*(3*a^2*b - b^3)*(f*x + e)*d*
cos(4*f*x + 4*e) - 4*(3*a^2*b - b^3)*(f*x + e)*d*cos(2*f*x + 2*e) + (6*I*a^2*b - 2*I*b^3)*(f*x + e)*d*sin(4*f*
x + 4*e) + (-12*I*a^2*b + 4*I*b^3)*(f*x + e)*d*sin(2*f*x + 2*e) + 2*(3*a^2*b - b^3)*(f*x + e)*d)*arctan2(sin(f
*x + e), -cos(f*x + e) + 1) + ((3*a^2*b - 3*I*a*b^2 - b^3)*(f*x + e)^2*d + (12*a*b^2*d + (6*I*a*b^2 + 2*b^3)*d
*e + (-6*I*a*b^2 - 2*b^3)*c*f)*(f*x + e))*cos(4*f*x + 4*e) - ((6*a^2*b - 6*I*a*b^2 - 2*b^3)*(f*x + e)^2*d - 2*
b^3*d + 4*(3*a*b^2 + I*b^3)*d*e - 4*(3*a*b^2 + I*b^3)*c*f - ((-12*I*a*b^2 - 4*b^3)*d*e + (12*I*a*b^2 + 4*b^3)*
c*f - 4*(3*a*b^2 - I*b^3)*d)*(f*x + e))*cos(2*f*x + 2*e) + (2*(3*a^2*b - b^3)*d*cos(4*f*x + 4*e) - 4*(3*a^2*b
- b^3)*d*cos(2*f*x + 2*e) + (6*I*a^2*b - 2*I*b^3)*d*sin(4*f*x + 4*e) + (-12*I*a^2*b + 4*I*b^3)*d*sin(2*f*x + 2
*e) + 2*(3*a^2*b - b^3)*d)*dilog(-e^(I*f*x + I*e)) + (2*(3*a^2*b - b^3)*d*cos(4*f*x + 4*e) - 4*(3*a^2*b - b^3)
*d*cos(2*f*x + 2*e) + (6*I*a^2*b - 2*I*b^3)*d*sin(4*f*x + 4*e) + (-12*I*a^2*b + 4*I*b^3)*d*sin(2*f*x + 2*e) +
2*(3*a^2*b - b^3)*d)*dilog(e^(I*f*x + I*e)) + (I*b^3*d*e - I*b^3*c*f + 3*I*a*b^2*d + (3*I*a^2*b - I*b^3)*(f*x
+ e)*d + (I*b^3*d*e - I*b^3*c*f + 3*I*a*b^2*d + (3*I*a^2*b - I*b^3)*(f*x + e)*d)*cos(4*f*x + 4*e) + (-2*I*b^3*
d*e + 2*I*b^3*c*f - 6*I*a*b^2*d + (-6*I*a^2*b + 2*I*b^3)*(f*x + e)*d)*cos(2*f*x + 2*e) - (b^3*d*e - b^3*c*f +
3*a*b^2*d + (3*a^2*b - b^3)*(f*x + e)*d)*sin(4*f*x + 4*e) + 2*(b^3*d*e - b^3*c*f + 3*a*b^2*d + (3*a^2*b - b^3)
*(f*x + e)*d)*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) + (I*b^3*d*e - I*b^3
*c*f + 3*I*a*b^2*d + (3*I*a^2*b - I*b^3)*(f*x + e)*d + (I*b^3*d*e - I*b^3*c*f + 3*I*a*b^2*d + (3*I*a^2*b - I*b
^3)*(f*x + e)*d)*cos(4*f*x + 4*e) + (-2*I*b^3*d*e + 2*I*b^3*c*f - 6*I*a*b^2*d + (-6*I*a^2*b + 2*I*b^3)*(f*x +
e)*d)*cos(2*f*x + 2*e) - (b^3*d*e - b^3*c*f + 3*a*b^2*d + (3*a^2*b - b^3)*(f*x + e)*d)*sin(4*f*x + 4*e) + 2*(b
^3*d*e - b^3*c*f + 3*a*b^2*d + (3*a^2*b - b^3)*(f*x + e)*d)*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e
)^2 - 2*cos(f*x + e) + 1) + ((3*I*a^2*b + 3*a*b^2 - I*b^3)*(f*x + e)^2*d - 2*(-6*I*a*b^2*d + (3*a*b^2 - I*b^3)
*d*e - (3*a*b^2 - I*b^3)*c*f)*(f*x + e))*sin(4*f*x + 4*e) + ((-6*I*a^2*b - 6*a*b^2 + 2*I*b^3)*(f*x + e)^2*d +
2*I*b^3*d + (-12*I*a*b^2 + 4*b^3)*d*e + (12*I*a*b^2 - 4*b^3)*c*f + (4*(3*a*b^2 - I*b^3)*d*e - 4*(3*a*b^2 - I*b
^3)*c*f + (-12*I*a*b^2 - 4*b^3)*d)*(f*x + e))*sin(2*f*x + 2*e))/(-2*I*f*cos(4*f*x + 4*e) + 4*I*f*cos(2*f*x + 2
*e) + 2*f*sin(4*f*x + 4*e) - 4*f*sin(2*f*x + 2*e) - 2*I*f))/f

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Fricas [B]  time = 1.93986, size = 1596, normalized size = 5.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*(a+b*cot(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/4*(2*(a^3 - 3*a*b^2)*d*f^2*x^2 - 4*b^3*c*f - 4*(b^3*d*f - (a^3 - 3*a*b^2)*c*f^2)*x - 2*((a^3 - 3*a*b^2)*d*f
^2*x^2 + 2*(a^3 - 3*a*b^2)*c*f^2*x)*cos(2*f*x + 2*e) - (-I*(3*a^2*b - b^3)*d*cos(2*f*x + 2*e) + I*(3*a^2*b - b
^3)*d)*dilog(cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) - (I*(3*a^2*b - b^3)*d*cos(2*f*x + 2*e) - I*(3*a^2*b - b^3
)*d)*dilog(cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e)) + 2*(3*a*b^2*d - (3*a^2*b - b^3)*d*e + (3*a^2*b - b^3)*c*f -
(3*a*b^2*d - (3*a^2*b - b^3)*d*e + (3*a^2*b - b^3)*c*f)*cos(2*f*x + 2*e))*log(-1/2*cos(2*f*x + 2*e) + 1/2*I*s
in(2*f*x + 2*e) + 1/2) + 2*(3*a*b^2*d - (3*a^2*b - b^3)*d*e + (3*a^2*b - b^3)*c*f - (3*a*b^2*d - (3*a^2*b - b^
3)*d*e + (3*a^2*b - b^3)*c*f)*cos(2*f*x + 2*e))*log(-1/2*cos(2*f*x + 2*e) - 1/2*I*sin(2*f*x + 2*e) + 1/2) + 2*
((3*a^2*b - b^3)*d*f*x + (3*a^2*b - b^3)*d*e - ((3*a^2*b - b^3)*d*f*x + (3*a^2*b - b^3)*d*e)*cos(2*f*x + 2*e))
*log(-cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e) + 1) + 2*((3*a^2*b - b^3)*d*f*x + (3*a^2*b - b^3)*d*e - ((3*a^2*b
- b^3)*d*f*x + (3*a^2*b - b^3)*d*e)*cos(2*f*x + 2*e))*log(-cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e) + 1) - 2*(6*a
*b^2*d*f*x + 6*a*b^2*c*f + b^3*d)*sin(2*f*x + 2*e))/(f^2*cos(2*f*x + 2*e) - f^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \cot{\left (e + f x \right )}\right )^{3} \left (c + d x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*(a+b*cot(f*x+e))**3,x)

[Out]

Integral((a + b*cot(e + f*x))**3*(c + d*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}{\left (b \cot \left (f x + e\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*(a+b*cot(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((d*x + c)*(b*cot(f*x + e) + a)^3, x)